Yohann De Castro

A rice method proof of the null-space property over the Grassmannian

The Null-Space Property (NSP) is a necessary and sufficient condition for the recovery of the largest coefficients of solutions to an under-determined system of linear equations. Interestingly, this property governs also the success and the failure of recent developments in high-dimensional statistics, signal processing, error-correcting codes and the theory of polytopes. Although this property is the keystone of

Optimal Designs for Lasso and Dantzig Selector Using Expander Codes

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Power of the spacing test for least-angle regression

-minimization techniques, it is an open problem to derive a closed form for the phase transition on NSP. In this article, we provide the first proof of NSP using random processes theory and the Rice method. As a matter of fact, our analysis gives non-asymptotic bounds for NSP with respect to unitarily invariant distributions. Furthermore, we derive a simple sufficient condition for NSP.

Spike detection from inaccurate samplings

We investigate the high-dimensional regression problem using adjacency matrices of unbalanced expander graphs. In this frame, we prove that the ℓ2-prediction error and ℓ1-risk of the lasso, and the Dantzig selector are optimal up to an explicit multiplicative constant. Thus, we can estimate a high-dimensional target vector with an error term similar to the one obtained in a situation where one knows the support of the largest coordinates in advance. Moreover, we show that these design matrices have an explicit restricted eigenvalue. Precisely, they satisfy the restricted eigenvalue assumption and compatibility condition with an explicit constant. Eventually, we capitalize on the recent construction of unbalanced expander graphs due to Guruswami, Umans, and Vadhan, to provide a deterministic polynomial time construction of these design matrices.

Exact Solutions to Super Resolution on Semi-Algebraic Domains in Higher Dimensions

Recent advances in Post-Selection Inference have shown that conditional testing is relevant and tractable in high-dimensions. In the Gaussian linear model, further works have derived unconditional test statistics such as the Kac-Rice Pivot for general penalized problems. In order to test the global null, a prominent offspring of this breakthrough is the spacing test that accounts the relative separation between the first two knots of the celebrated least-angle regression (LARS) algorithm. However, no results have been shown regarding the distribution of these test statistics under the alternative. For the first time, this paper addresses this important issue for the spacing test and shows that it is unconditionally unbiased. Furthermore, we provide the first extension of the spacing test to the frame of unknown noise variance. More precisely, we investigate the power of the spacing test for LARS and prove that it is unbiased: its power is always greater or equal to the significance level

A REMARK ON THE LASSO AND THE DANTZIG SELECTOR

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Minimax adaptive estimation of nonparametric hidden Markov models

. In particular, we describe the power of this test under various scenarii: we prove that its rejection region is optimal when the predictors are orthogonal; as the level

Non-uniform spline recovery from small degree polynomial approximation

\alpha

Reconstructing undirected graphs from eigenspaces

goes to zero, we show that the probability of getting a true positive is much greater than

Estimating the transition matrix of a Markov chain observed at random times

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